Shift invariant spaces on compact groups

We study the theory of shift invariant spaces in L2(G), where G is a compact group. We define a range function and show that a relation between an H-invariant space and the range function is valid as in the case of abelian group setting. Here H is assumed to be a closed normal subgroup of G. We also obtain a decomposition for an H-invariant space in terms of principle H-invariant spaces whose generators give rise to “generalized Parseval frames” and use this result to study H-preserving operators.